There are applied power mappings in algebras with logarithms induced
by a given linear operator D in order to study particular properties of powers
of logarithms. Main results of this paper will be concerned with the case
when an algebra under consideration is commutative and has a unit and
the operator D satisfies the Leibniz condition, i.e. D(xy) = xDy + yDx for
x, y ∈ dom D. Note that in the Number Theory there are well-known several
formulae expressed by means of some combinations of powers of logarithmic
and antilogarithmic mappings or powers of logarithms and antilogarithms
(cf. for instance, the survey of Schinzel S[1].